New York City Taxi and Limousine commission Data Analysis
Nana Akwasi Abayie Boateng
The Code below loads all the packages that would be required for this project.If a package is not available in the library,it will be downloaded and loaded.
#==================================================================
# The follwing code :
# 1 Load all packages required for the analysis at the same time
# 2 Check if a required package is not already installed.
#3 If a required package is not already installed, the install thatt package
#
#==================================================================
list=c("tidyverse","stringr","forcats","ggmap","rvest","tm","SnowballC","dplyr","calibrate","doParallel",
"stringi","ggplot2","maps","httr","rsdmx","devtools","plyr","dplyr","ggplot2","caret","elasticnet",
"magrittr","broom","glmnet","Hmisc",'knitr',"RSQLite","RANN","lubridate","ggvis","plotly","lars",
"ggcorrplot","GGally","ROCR","lattice","car","corrgram","ggcorrplot","sqldf","parallel")
list_packages <- list
new.packages <- list_packages[!(list_packages %in% installed.packages()[,"Package"])]
if(length(new.packages)) install.packages(new.packages)
sapply(list, require, character.only = TRUE)Loading required package: tidyverse
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package <U+393C><U+3E31>dplyr<U+393C><U+3E32> was built under R version 3.3.3Conflicts with tidy packages -------------------------------------------------------------------------------------------
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Google Maps API Terms of Service: http://developers.google.com/maps/terms.
Please cite ggmap if you use it: see citation('ggmap') for details.
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package <U+393C><U+3E31>doParallel<U+393C><U+3E32> was built under R version 3.3.3Loading required package: foreach
foreach: simple, scalable parallel programming from Revolution Analytics
Use Revolution R for scalability, fault tolerance and more.
http://www.revolutionanalytics.com
Attaching package: <U+393C><U+3E31>foreach<U+393C><U+3E32>
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Loading required package: plyr
----------------------------------------------------------------------------------------------------------------------
You have loaded plyr after dplyr - this is likely to cause problems.
If you need functions from both plyr and dplyr, please load plyr first, then dplyr:
library(plyr); library(dplyr)
----------------------------------------------------------------------------------------------------------------------
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TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
stringi ggplot2 maps httr rsdmx devtools plyr dplyr ggplot2 caret
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parallel
TRUE #==================================================================
# Set up parallel processing
# leave two cores for operating system
#==================================================================
cluster <- makeCluster(detectCores() - 2)
registerDoParallel(cluster)The September 2015 data green taxi contains 1048575 rows and 21 columns.
#==================================================================
# load the downloaded data with the readr package
# dim function produces the dimension of the data
# the September 2015 data contains 1048575 rows and 21 columns
#==================================================================
data=readr::read_csv("C:/Users/Gucci148/Documents/DataMiningscience/Capitalone/green_tripdata_2015-09.csv")Parsed with column specification:
cols(
.default = col_double(),
VendorID = col_integer(),
lpep_pickup_datetime = col_character(),
Lpep_dropoff_datetime = col_character(),
Store_and_fwd_flag = col_character(),
RateCodeID = col_integer(),
Passenger_count = col_integer(),
Ehail_fee = col_character(),
Payment_type = col_integer(),
Trip_type = col_integer()
)
See spec(...) for full column specifications.
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|================================================================================| 100% 122 MBnames(data) [1] "VendorID" "lpep_pickup_datetime" "Lpep_dropoff_datetime" "Store_and_fwd_flag"
[5] "RateCodeID" "Pickup_longitude" "Pickup_latitude" "Dropoff_longitude"
[9] "Dropoff_latitude" "Passenger_count" "Trip_distance" "Fare_amount"
[13] "Extra" "MTA_tax" "Tip_amount" "Tolls_amount"
[17] "Ehail_fee" "improvement_surcharge" "Total_amount" "Payment_type"
[21] "Trip_type" dim(data)[1] 1048575 21#==================================================================
# Look at the structure of the data with the glimpse function in
# dplyr package
#==================================================================
dplyr::glimpse(data)Observations: 1,048,575
Variables: 21
$ VendorID <int> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2...
$ lpep_pickup_datetime <chr> "9/1/2015 0:02", "9/1/2015 0:04", "9/1/2015 0:01", "9/1/2015 0:02", "9/1/2015 0:00", ...
$ Lpep_dropoff_datetime <chr> "9/1/2015 0:02", "9/1/2015 0:04", "9/1/2015 0:04", "9/1/2015 0:06", "9/1/2015 0:04", ...
$ Store_and_fwd_flag <chr> "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", ...
$ RateCodeID <int> 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
$ Pickup_longitude <dbl> -73.97948, -74.01080, -73.92141, -73.92139, -73.95548, -73.94530, -73.89088, -73.9467...
$ Pickup_latitude <dbl> 40.68496, 40.91222, 40.76671, 40.76668, 40.71405, 40.80819, 40.74643, 40.79732, 40.69...
$ Dropoff_longitude <dbl> -73.97943, -74.01078, -73.91441, -73.93143, -73.94441, -73.93767, -73.87692, -73.9376...
$ Dropoff_latitude <dbl> 40.68502, 40.91221, 40.76469, 40.77158, 40.71473, 40.82120, 40.75631, 40.80452, 40.68...
$ Passenger_count <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 6, 1, 1, 1, 1, 1, 1...
$ Trip_distance <dbl> 0.00, 0.00, 0.59, 0.74, 0.61, 1.07, 1.43, 0.90, 1.33, 0.84, 0.80, 0.70, 1.01, 0.39, 0...
$ Fare_amount <dbl> 7.8, 45.0, 4.0, 5.0, 5.0, 5.5, 6.5, 5.0, 6.0, 5.5, 5.0, 4.0, 5.5, 3.5, 4.0, 7.5, 7.5,...
$ Extra <dbl> 0.0, 0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, ...
$ MTA_tax <dbl> 0.0, 0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, ...
$ Tip_amount <dbl> 1.95, 0.00, 0.50, 0.00, 0.00, 1.36, 0.00, 0.00, 1.46, 0.00, 0.00, 1.06, 0.00, 0.00, 0...
$ Tolls_amount <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
$ Ehail_fee <chr> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N...
$ improvement_surcharge <dbl> 0.0, 0.0, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3, ...
$ Total_amount <dbl> 9.75, 45.00, 5.80, 6.30, 6.30, 8.16, 7.80, 6.30, 8.76, 6.80, 6.30, 6.36, 6.80, 4.80, ...
$ Payment_type <int> 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2...
$ Trip_type <int> 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...summary(data) VendorID lpep_pickup_datetime Lpep_dropoff_datetime Store_and_fwd_flag RateCodeID Pickup_longitude
Min. :1.000 Length:1048575 Length:1048575 Length:1048575 Min. : 1.000 Min. :-83.32
1st Qu.:2.000 Class :character Class :character Class :character 1st Qu.: 1.000 1st Qu.:-73.96
Median :2.000 Mode :character Mode :character Mode :character Median : 1.000 Median :-73.95
Mean :1.784 Mean : 1.099 Mean :-73.83
3rd Qu.:2.000 3rd Qu.: 1.000 3rd Qu.:-73.92
Max. :2.000 Max. :99.000 Max. : 0.00
Pickup_latitude Dropoff_longitude Dropoff_latitude Passenger_count Trip_distance Fare_amount
Min. : 0.00 Min. :-83.43 Min. : 0.00 Min. :0.000 Min. : 0.000 Min. :-475.00
1st Qu.:40.70 1st Qu.:-73.97 1st Qu.:40.70 1st Qu.:1.000 1st Qu.: 1.100 1st Qu.: 6.50
Median :40.75 Median :-73.94 Median :40.75 Median :1.000 Median : 2.000 Median : 9.50
Mean :40.69 Mean :-73.84 Mean :40.69 Mean :1.375 Mean : 2.997 Mean : 12.66
3rd Qu.:40.80 3rd Qu.:-73.91 3rd Qu.:40.79 3rd Qu.:1.000 3rd Qu.: 3.800 3rd Qu.: 15.50
Max. :43.18 Max. : 0.00 Max. :42.80 Max. :9.000 Max. :246.280 Max. : 580.50
Extra MTA_tax Tip_amount Tolls_amount Ehail_fee improvement_surcharge
Min. :-1.0000 Min. :-0.5000 Min. :-12.300 Min. :-5.5400 Length:1048575 Min. :-0.3000
1st Qu.: 0.0000 1st Qu.: 0.5000 1st Qu.: 0.000 1st Qu.: 0.0000 Class :character 1st Qu.: 0.3000
Median : 0.5000 Median : 0.5000 Median : 0.000 Median : 0.0000 Mode :character Median : 0.3000
Mean : 0.3396 Mean : 0.4865 Mean : 1.235 Mean : 0.1282 Mean : 0.2919
3rd Qu.: 0.5000 3rd Qu.: 0.5000 3rd Qu.: 2.000 3rd Qu.: 0.0000 3rd Qu.: 0.3000
Max. :12.0000 Max. : 0.5000 Max. :300.000 Max. :95.7500 Max. : 0.3000
Total_amount Payment_type Trip_type
Min. :-475.00 Min. :1.000 Min. :1.000
1st Qu.: 8.16 1st Qu.:1.000 1st Qu.:1.000
Median : 11.80 Median :2.000 Median :1.000
Mean : 15.14 Mean :1.545 Mean :1.023
3rd Qu.: 18.35 3rd Qu.:2.000 3rd Qu.:1.000
Max. : 581.30 Max. :5.000 Max. :2.000
NA's :3 #==================================================================
#
# 1 check missing obervations of each column variable
# 2 sum of all missing obervations in data
# 3 Ehail_fee has all rows missing,remove from dataset
#==================================================================
apply(is.na.data.frame(data),2,sum) VendorID lpep_pickup_datetime Lpep_dropoff_datetime Store_and_fwd_flag RateCodeID
0 0 0 0 0
Pickup_longitude Pickup_latitude Dropoff_longitude Dropoff_latitude Passenger_count
0 0 0 0 0
Trip_distance Fare_amount Extra MTA_tax Tip_amount
0 0 0 0 0
Tolls_amount Ehail_fee improvement_surcharge Total_amount Payment_type
0 1048575 0 0 0
Trip_type
3 newdata=dplyr::select(data,-Ehail_fee)
sum(is.na(newdata))[1] 3The average elapsed trip distance reported by the taximeter is approximately 3 miles The median elapsed trip distance reported by the taximeter is approximately 2 miles. The highest elapsed trip distance reported by the taximeter is approximately 603 miles and lowest is 0. The 603 mile appears to be an outlier. The data is skewed to the right with most observations are between zero and four.
#==================================================================
#
# Histogram of the Distribution of Trip Distance
# with ggplot2 package
#==================================================================
p<-ggplot(newdata, aes(x =Trip_distance )) +
geom_histogram(fill="black",col="black",alpha=0.2,binwidth=0.02) +
theme_minimal() + ggtitle(" Histogram of the Distribution of Trip Distance") +
#center title
theme(plot.title = element_text(hjust = 0.5)) +xlab("Trip Distance")+ylab("Frequency")
ggplotly(p)Right skewed distributions are not symmetric and transformations is usually done in order to build any model that uses a normal distribution assumption.Transformations improve symmetry,linear relationships and constant variance which are usually need for models such as regression. A common transformation of right skewed data is the log transformation.
#==================================================================
#
# Histogram of the Distribution of Log Trip Distance
# with ggplot2 package
#==================================================================
newdata=newdata%>%mutate(New_distance=log(Trip_distance+1))
p=ggplot(newdata, aes(x =New_distance ))+ geom_histogram(fill="black",col="black",binwidth=0.5) +
theme_minimal() + ggtitle(" Histogram of the Distribution of Trip Distance") +
#center title
theme(plot.title = element_text(hjust = 0.5)) +xlab("Log of Trip Distance")+ylab("Frequency")
ggplotly(p)The summary statistics of the transformed Trip distance shows the mean is about 3.027 and median is 2.010.
newdata%>%mutate(Transform=log(Trip_distance+1))%>%dplyr::select(Trip_distance)%>%summary%>%knitr::kable()| Trip_distance | |
|---|---|
| Min. : 0.000 | |
| 1st Qu.: 1.100 | |
| Median : 2.000 | |
| Mean : 2.997 | |
| 3rd Qu.: 3.800 | |
| Max. :246.280 |
The mean and median trip distance grouped by hour of day. The maximum mean and median trip distance by hour which is 4.192 and 2.95 respectively was travelled at fifth hour.
#===========================================================================
#
# Mean and median trip distance grouped by hour of day.
# Create Hours variable by extracting hours from the
# the date and time when the meter was engaged (lpep_pickup_datetime).
#===========================================================================
newdata=newdata%>%dplyr::mutate(Hours= format(as.POSIXct(strptime(newdata$lpep_pickup_datetime,"%m/%d/%Y %H:%M",tz="")) ,format = "%H"))
newdata%>% mutate(Hours=as.numeric(Hours))%>%group_by(Hours)%>% dplyr::summarise(Mean=mean(Trip_distance),Median=median(Trip_distance),n=n())#===========================================================================
# Find the Hour in which the maximum mean Trip distance occured
#===========================================================================
newdata%>% mutate(Hours=as.numeric(Hours))%>%group_by(Hours)%>% dplyr::summarise(Mean=mean(Trip_distance),Median=median(Trip_distance),n=n())%>%
dplyr::slice(which.max(Mean )) #===========================================================================
# Find the Hour in which the maximum median Trip distance occured
#===========================================================================
newdata%>% mutate(Hours=as.numeric(Hours))%>%group_by(Hours)%>% dplyr::summarise(Mean=mean(Trip_distance),Median=median(Trip_distance),n=n())%>%
dplyr::slice(which.max(Median )) The new variable Tip is created by dividing Tip amount by the Total amount.The minimum Tip amount is -12.30. This appears to be an error in the data. The Tip amounts distribution is also right skewed with most observations between 0 and 10. (99%).
#==================================================================
#
# derived variable for tip as a percentage of the total fare
#==================================================================
newdata=newdata%>%dplyr::mutate(Tip=(Tip_amount/Total_amount)*100)
length(newdata$Tip_amount[newdata$Tip_amount<10])/length(newdata$Tip_amount)[1] 0.9917707#==================================================================
#
# Histogram of the Distribution of Tip_amount
# with ggplot2 package
#==================================================================
p<-ggplot2::ggplot(newdata, aes(x =Tip_amount )) +
geom_histogram(fill="black",col="black",alpha=0.2,binwidth=0.02) +
theme_minimal() + ggtitle(" Histogram of the Distribution of Tip amount") +
#center title
theme(plot.title = element_text(hjust = 0.5)) +xlab("Tip_amount")+ylab("Frequency")
ggplotly(p)# log transformation ,13.3 is added to prevent a case of taking log(0)
newdata=newdata%>%dplyr::mutate(New_Tip=log(Tip_amount+13.3))
The log transformation did not help out much in moving the data close to normal.We keep the old form in further analysis.
p=ggplot2::ggplot(newdata, aes(x =New_Tip )) +
geom_histogram(fill="black",col="black",alpha=0.2,binwidth=0.2) +
theme_minimal() + ggtitle(" Histogram of the Distribution of New Tip amount") +
#center title
theme(plot.title = element_text(hjust = 0.5)) +xlab("New Tip amount")+ylab("Frequency")
ggplotly(p)#==============================================================================
#
# Building a Predictive Model
# Preprocessing data
# 1 There are three missing variables that will be removed at preprocessing step
# 2 Center and scale numerical columns to reduce variability
#==============================================================================
newdata=newdata[complete.cases(newdata),]
sum(is.na(newdata))[1] 0summary(newdata) VendorID lpep_pickup_datetime Lpep_dropoff_datetime Store_and_fwd_flag RateCodeID Pickup_longitude
Min. :1.000 Length:1045608 Length:1045608 Length:1045608 Min. :1.000 Min. :-83.32
1st Qu.:2.000 Class :character Class :character Class :character 1st Qu.:1.000 1st Qu.:-73.96
Median :2.000 Mode :character Mode :character Mode :character Median :1.000 Median :-73.95
Mean :1.785 Mean :1.089 Mean :-73.84
3rd Qu.:2.000 3rd Qu.:1.000 3rd Qu.:-73.92
Max. :2.000 Max. :6.000 Max. : 0.00
Pickup_latitude Dropoff_longitude Dropoff_latitude Passenger_count Trip_distance Fare_amount
Min. : 0.00 Min. :-83.43 Min. : 0.00 Min. :0.000 Min. : 0.000 Min. :-475.0
1st Qu.:40.70 1st Qu.:-73.97 1st Qu.:40.70 1st Qu.:1.000 1st Qu.: 1.100 1st Qu.: 6.5
Median :40.75 Median :-73.95 Median :40.75 Median :1.000 Median : 2.000 Median : 10.0
Mean :40.69 Mean :-73.85 Mean :40.70 Mean :1.376 Mean : 2.997 Mean : 12.7
3rd Qu.:40.80 3rd Qu.:-73.91 3rd Qu.:40.79 3rd Qu.:1.000 3rd Qu.: 3.800 3rd Qu.: 15.5
Max. :43.18 Max. : 0.00 Max. :42.80 Max. :9.000 Max. :246.280 Max. : 580.5
Extra MTA_tax Tip_amount Tolls_amount improvement_surcharge Total_amount
Min. :-1.0000 Min. :-0.5000 Min. :-12.300 Min. :-5.5400 Min. :-0.3000 Min. :-475.00
1st Qu.: 0.0000 1st Qu.: 0.5000 1st Qu.: 0.000 1st Qu.: 0.0000 1st Qu.: 0.3000 1st Qu.: 8.16
Median : 0.5000 Median : 0.5000 Median : 0.000 Median : 0.0000 Median : 0.3000 Median : 11.80
Mean : 0.3405 Mean : 0.4878 Mean : 1.238 Mean : 0.1286 Mean : 0.2928 Mean : 15.18
3rd Qu.: 0.5000 3rd Qu.: 0.5000 3rd Qu.: 2.000 3rd Qu.: 0.0000 3rd Qu.: 0.3000 3rd Qu.: 18.36
Max. :12.0000 Max. : 0.5000 Max. :300.000 Max. :95.7500 Max. : 0.3000 Max. : 581.30
Payment_type Trip_type Hours Tip
Min. :1.000 Min. :1.00 Length:1045608 Min. : 0.000
1st Qu.:1.000 1st Qu.:1.00 Class :character 1st Qu.: 0.000
Median :2.000 Median :1.00 Mode :character Median : 0.000
Mean :1.544 Mean :1.02 Mean : 6.573
3rd Qu.:2.000 3rd Qu.:1.00 3rd Qu.: 16.667
Max. :5.000 Max. :2.00 Max. :100.000 #==================================================================
#Spliting training set into two parts based on outcome:
# Training set which 70% and Test set which 30% of the data
#==================================================================
index <- createDataPartition(newdata$Tip_amount, p=0.70, list=FALSE)
trainSet <- newdata[ index,]
testSet <- newdata[-index,]
sum(is.na(trainSet))[1] 0sum(is.na(testSet))[1] 0sum(is.na(newdata))[1] 0There are very high positive correlations between Total amount and Trip distance,of course which makes sense Total amount and Fare amount that passengers who travelled longer distances were charged more. There is also a significant positive correlation between Total amount and Tip amount. Passengers who travelled longer distances and paid higher fares were more likely to tip.
#==================================================================
# Exploratory data analysis
# Determine correlation between variables
#==================================================================
newdata%>%dplyr::select(New_distance,Fare_amount,improvement_surcharge,Tolls_amount,Tip_amount,MTA_tax
,Extra,Passenger_count,VendorID, Tip,Total_amount)%>%cor New_distance Fare_amount improvement_surcharge Tolls_amount Tip_amount MTA_tax
New_distance 1.000000000 0.782704516 0.061096337 0.253663550 0.363260074 0.05577728
Fare_amount 0.782704516 1.000000000 -0.053465833 0.284537088 0.395533646 -0.07284627
improvement_surcharge 0.061096337 -0.053465833 1.000000000 -0.001387602 0.020457209 0.96242103
Tolls_amount 0.253663550 0.284537088 -0.001387602 1.000000000 0.190925008 -0.02705827
Tip_amount 0.363260074 0.395533646 0.020457209 0.190925008 1.000000000 0.01258670
MTA_tax 0.055777284 -0.072846266 0.962421031 -0.027058271 0.012586701 1.00000000
Extra -0.019493758 -0.050765468 0.153176317 -0.030179026 -0.006751906 0.15466893
Passenger_count 0.015683087 0.012817630 0.001881064 0.008503446 0.005471641 0.00137572
VendorID 0.004464695 0.007812296 -0.015807901 0.006219987 -0.007280304 -0.02376706
Tip 0.121205434 0.087938246 0.063133360 0.044409632 0.720463834 0.06393077
Total_amount 0.779603050 0.976113587 -0.026739200 0.365396798 0.571074413 -0.04716232
Extra Passenger_count VendorID Tip Total_amount
New_distance -0.019493758 0.015683087 0.004464695 0.121205434 0.779603050
Fare_amount -0.050765468 0.012817630 0.007812296 0.087938246 0.976113587
improvement_surcharge 0.153176317 0.001881064 -0.015807901 0.063133360 -0.026739200
Tolls_amount -0.030179026 0.008503446 0.006219987 0.044409632 0.365396798
Tip_amount -0.006751906 0.005471641 -0.007280304 0.720463834 0.571074413
MTA_tax 0.154668930 0.001375720 -0.023767065 0.063930768 -0.047162317
Extra 1.000000000 0.020607742 0.006560547 0.011569255 -0.015422984
Passenger_count 0.020607742 1.000000000 0.080734881 0.001885730 0.013657879
VendorID 0.006560547 0.080734881 1.000000000 -0.008829697 0.005745142
Tip 0.011569255 0.001885730 -0.008829697 1.000000000 0.233311439
Total_amount -0.015422984 0.013657879 0.005745142 0.233311439 1.000000000ggplot(trainSet , aes(Tip, Tip_amount))+geom_point(color="purple")+
ggtitle('Tip vs Tip_amount')+
xlab('Tip')+ylab('Tip_amount')+
stat_smooth(method=lm, colour='black',span=0.2)
There exist a strong association between Tip and Tip amount.
ggplot(trainSet , aes(Tip, Fare_amount))+geom_point(color="purple")+
ggtitle('Tip vs Fare_amount')+
xlab('Tip')+ylab('Fare_amount')+
stat_smooth(method=lm, colour='black',span=0.2)
ggplot(trainSet , aes(Tip, Trip_distance))+geom_point(color="purple")+
ggtitle('Tip vs Trip_distance')+
xlab('Tip')+ylab('Trip_distance')+
stat_smooth(method=lm, colour='black',span=0.2)
The figure shows there are a few outlier points ,where the Tip was low for a very long Trip distance
ggplot(trainSet , aes(Tip, New_distance))+geom_point(color="purple")+
ggtitle('Tip vs New_distance')+
xlab('Tip')+ylab('New_distance')+
stat_smooth(method=lm, colour='black',span=0.2)
ggplot(trainSet , aes(Tip, Tolls_amount))+geom_point(color="purple")+
ggtitle('Tip vs Tolls_amount')+
xlab('Tip')+ylab('Tolls_amount')+
stat_smooth(method=lm, colour='black',span=0.2)
The Figure does reveal a weak positive association between Tolls amount and Tip given.
#==================================================================
# Do feature selection using Generalized Linear Model
#==================================================================
control <- trainControl(method="repeatedcv", number=10, repeats=5)
my_glm<- train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax","improvement_surcharge",
"Extra","Passenger_count","VendorID")], trainSet$Tip,
method = "glm",
preProc = c("center", "scale"),
trControl = control)Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.Setting row names on a tibble is deprecated.ggplot(varImp(my_glm, scale = FALSE))+ggtitle("Variable Importance using GLM")+theme_minimal()
From the variable importance plot, we drop can drop improvement surcharge since it has have very little predictive power for Tip.
#===============================================================================
# Build a Generalized Linear Model For with Stepwise Selection prediction
#===============================================================================
control <- trainControl(method="repeatedcv", number=10, repeats=5)
step_glm<- train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","Passenger_count","VendorID")], trainSet$Tip,
method = "glmStepAIC",
preProc = c("center", "scale"),
trControl = control)pred<-predict(step_glm,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")])Error in predict(step_glm, testSet[, c("Fare_amount", "Trip_distance", :
object 'step_glm' not found# Print, plot variable importance
print(varImp(my_glm, scale = FALSE))
plot(varImp(my_glm, scale = FALSE), main="Variable Importance using GLM")
# Mean Squared Error
sqrt(mean(pred- testSet$Tip)^2)#===============================================================================
# Build a Partial Least Squares Model
#===============================================================================
# Partial Least Squares is one way to reduce dimension of the predictors used in the model. It identifies
#linear combinations,or directions, that best represent the predictors in the data.The directions are identified in
# unsupervised way since the outcome variable is not in identifying the principal
# component directions.The predictors are preprocessed by centering and scaling.
# PLS will seek
#directions of maximum variation while simultaneously considering correlation with the response.
control <- trainControl(method="repeatedcv", number=10, repeats=5)
plsfit=train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID","Passenger_count")], trainSet$Tip,
method = "glmStepAIC",tuneLength = 20,
preProc = c("center", "scale"),
trControl = control)
# Predict using test data
pred<-predict(plsfit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")])
my_data=data_frame(predicted=pred,observed=testSet$Tip)
ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('PLS')
# Print, plot variable importance
print(varImp(plsfit, scale = FALSE))
plot(varImp(plsfit, scale = FALSE), main="Variable Importance using PLS")# Mean Squared Error
sqrt(mean(pred- testSet$Tip)^2)#===============================================================================
# Build Penalized linear regression models Model
# Lasso (least absolute shrinkage and selection operator)
#===============================================================================
# The LASSO penalizes the model for having many predictors by shrinking the coefficients of
#
# those predictors with little variation to zero thereby reducing the dimension of the model.
lassoGrid <- expand.grid(.fraction = seq(.05, 1, length = 20))
control <- trainControl(method="repeatedcv", number=10, repeats=5)
set.seed(100)
lassofit=train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")], trainSet$Tip,
method = "lasso",tuneGrid = lassoGrid,
preProc = c("center", "scale"),
trControl = control)Setting row names on a tibble is deprecated.# Predict using test data
pred<-predict(lassofit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")])
my_data=data_frame(predicted=pred,observed=testSet$Tip)
ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('LASSO Model')
print(varImp(lassofit, scale = FALSE), main="Variable Importance using lasso")argument is not numeric or logical: returning NAError in model.frame.default(formula = y ~ x, na.action = na.omit, drop.unused.levels = TRUE) :
invalid type (list) for variable 'y'plot(varImp(lassofit, scale = FALSE), main="Variable Importance using lasso")argument is not numeric or logical: returning NAError in model.frame.default(formula = y ~ x, na.action = na.omit, drop.unused.levels = TRUE) :
invalid type (list) for variable 'y'plot(lassofit)
#Extract coefficients of final model
predict.enet(lassofit$finalModel, type='coefficients', s=lassofit$bestTune$fraction, mode='fraction')$s
[1] 1
$fraction
0
1
$mode
[1] "fraction"
$coefficients
Fare_amount Trip_distance Tolls_amount MTA_tax Extra VendorID
0.22877995 0.65357915 0.13175181 0.57739305 0.04522287 -0.07149811 # Mean Squared Error
sqrt(mean(pred- testSet$Tip)^2)[1] 0.0003035775I ran out memory after several attempts to fit a random forest model.
#===============================================================================
# Random Forest linear regression models Model
#
#===============================================================================
control <- trainControl(method="repeatedcv", number=10, repeats=5)
rf_fit<-train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")], trainSet$Tip,
method = "rf", ntrees = 1000,importance = TRUE,
preProc = c("center", "scale"),
trControl = control)
# Predict using the test data
pred<-predict(rf_fit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")])
my_data=adata_frame(predicted=pred,observed=testSet$Tip)
ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('Random Forest Model')
plot(varImp(rf_fit, scale = FALSE),main="Variable Importance using RF")# Mean Squared Error
sqrt(mean(pred- testSet$Tip)^2)I ran out memory after several attempts to fit a gradient boosting machine.
#===============================================================================
# Gradient Boosting Machine Model
#
#===============================================================================
# To tune over interaction depth, number of trees, and shrinkage first define a tuning grid,
#then train over this grid
gbmGrid <- expand.grid(.interaction.depth = c(2,5,8),
.n.trees = c(500, 1000,2000,5000),
.shrinkage = c(0.01, 0.1),
.n.minobsinnode=c(5,10,15))
control <- trainControl(method="repeatedcv", number=10, repeats=5)
gbm_fit<-train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")], trainSet$Tip,
method = "gbm",verbose=F,tuneGrid = gbmGrid,
preProc = c("center", "scale"),
trControl = control)
# Predict using the test data
pred<-predict(gbm_fit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
"Extra","VendorID")])
my_data=adata_frame(predicted=pred,observed=testSet$Tip)
ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('Boosting')
plot(varImp(gbm_fit, scale = FALSE),main="Variable Importance using Boosting")# Mean Squared Error
sqrt(mean(pred- testSet$Tip)^2)We constructed a derived variable of time difference by subtracting pick up time from drop off time.The average speed of any trip was obtained by dividing distance covered by the time take. The time is converted to seconds to hours which is standard form.The Overall average speed off all trips is about 13 miles per hour.
#Build a derived variable representing the average speed over the course of a trip.
#Average Speed=(distance)/time
# 1 create a TimeDifference variable by subtracting lpep_pickup_datetime
# from Lpep_dropoff_datetime using the difftime function.
#2 Extract the numeric values in difference in time(in seconds) by
# by the stri_extract_all_regex function
# 3 unlist the result which is characer and convert to numeric
# 4 convert time in seconds to hours by multiplying by 0.000277778
# This standardizes speed in scientific units which is miles per hour
# 5 Time difference greater than zero seconds were selected in computing the
# speed to avoid undefined mathematical expressions
# 6 Speed variable=Trip distance/ TimeDifference
# Average speed=mean of the Speed variable
# The Average speed over all of september is approximately 5.844 meters per second
newdata=newdata%>%mutate(TimeDifference=
as.numeric(unlist( stri_extract_all_regex(difftime(as.POSIXct(Lpep_dropoff_datetime, format="%m/%d/%Y %H:%M", tz="")
, as.POSIXct(lpep_pickup_datetime, format="%m/%d/%Y %H:%M", tz="")
, tz="",units = , "secs"), "[0-9]+"))))%>%mutate(Time=TimeDifference*0.000277778)
newdata=newdata[newdata$Time>0,]
#newdata=newdata%>%mutate(Speed=Trip_distance/TimeDifference)%>%dplyr::filter[newdata,TimeDifference >0]
# The filter approach seems not to recognise TimeDifference variable just created.
newdata=newdata%>%mutate(Speed=Trip_distance/Time)The overall average speed for all trips by green taxis is found to be 13.06 miles per hour.
newdata%>%dplyr::summarise(Average_Speed=mean(Speed))The average trip speeds are materially different in all weeks of September.We tested for the difference in means between the three weeks in September per the dataset by an using Kruskal-Wallis and ANOVA model.Since the trip distance is right skewed,we proceed further to use a non-parametric test like Kruskal-Wallis to test the differences between the speeds over the three weeks in September.Both parametric Anova without any transformations gives a similar outcome like the non-parametric Kruskal-Wallis test.We found thet there is asignificant difference in the average Speed by week travelled in September(p-value < 2.2e-16).The first week had the highest average speed of about 13.75. Labor weekend occurred in the first week of Septtember 2015,that could have contributed to the Higher speed in the first week of September as a lot of people were in a rush to get to their destinations For the holidays.
# Test of the mean and Median of average trip speeds
# in all weeks of September
# 1 Create a derived variable Week for the number of weeks in Septemeber
# present in the data
# 2 We then group speed by week
# 3 Test for difference in Mean speed for the weeks in September.
# 4 We will perform Kruskal Wallis nonparametric test and compare with parametric ANOVA
#test of the mean Speeds in the 3 weeks in September. The ANOVA test assumes the data is
# normally distributed wheras Kruskal-Wallis test does not.
week1=c("09/01/2015","09/02/2015","09/03/2015","09/04/2015","09/05/2015","09/06/2015","09/07/2015")
week2=c("09/08/2015","09/09/2015","09/10/2015","09/11/2015","09/12/2015","09/13/2015","09/14/2015")
week3=c("09/15/2015","09/16/2015","09/17/2015","09/18/2015","09/19/2015","09/20/2015","09/21/2015")
newdata=newdata%>%
dplyr::mutate(dates= format(as.POSIXct(strptime(newdata$lpep_pickup_datetime,"%m/%d/%Y %H:%M",tz="")) ,format = "%m/%d/%Y"))
newdata=newdata %>%
mutate(Week = case_when(.$dates %in% week1 ~ 1, .$dates %in% week2 ~ 2,.$dates %in% week3 ~ 3))
newdata# Find the mean speed in each in September
newdata %>%group_by(Week)%>% dplyr::summarise(mean=mean(Speed))NAnewdata %>%dplyr::select( Week,Speed)%>%mutate(Week=factor(Week))%>%kruskal.test()%>%tidy()some elements of 'x' are not numeric and will be coerced to numericnewdata %>%dplyr::select( Week,Speed)%>%lm(formula = Speed ~ factor(Week ))%>%anova()%>%tidy()A plausible Hypothesis would be to test if speed is associated with time of the day.This can be done by building a Linear regression model. There exist a significant association between Speed and Hour of the day. The coefficient of Hours is negative which indicates a negative relationship between the two variables. For every unit increase in Hour of the day,the Speed decreases by a factor of about 0.106343.
ggplot(newdata, aes(Hours, Speed))+geom_point(color="purple")+
ggtitle('Hours vs Speed')+
xlab('Hours')+ylab('Speed')+
stat_smooth(method=lm, colour='black',span=0.2)
newdata %>%dplyr::select(Speed,Hours)%>%mutate( Hours=as.numeric( Hours))%>%
lm(formula = Speed ~ (Hours ))%>%summary()
Call:
lm(formula = Speed ~ (Hours), data = .)
Residuals:
Min 1Q Median 3Q Max
-14.5 -3.7 -1.3 2.1 4157.8
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 14.489476 0.024647 587.88 <2e-16 ***
Hours -0.106343 0.001637 -64.94 <2e-16 ***
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Residual standard error: 11.42 on 1033531 degrees of freedom
Multiple R-squared: 0.004064, Adjusted R-squared: 0.004063
F-statistic: 4218 on 1 and 1033531 DF, p-value: < 2.2e-16fit_glm=glm( Speed ~ as.numeric(Hours), data=newdata)
tidy(fit_glm)head(augment(fit_glm))glance(fit_glm)Other insights one would consider given ample time include the following:
- Spatial data analysis with get_map and ggmap to determine if the green taxis followed the citys order to pick up paasengers from Manhattan eclusionary zone.
Building predictive models with deep neural networks which is known to have better predictive power.
Showing interactive plots to assist in detecting any underlying patterns of the data.
---
title: "New York City Taxi and Limousine commission Data Analysis"
output: html_notebook
---

## Nana Akwasi Abayie Boateng

The Code below loads all the packages that would be required for this project.If a package is not available
in the library,it will be downloaded and loaded.

```{r}
#==================================================================
#  The follwing code :
# 1  Load all packages required for the analysis at the same time
# 2  Check if a required package is not already installed.
#3   If a required package is not already installed, the install thatt package
# 
#==================================================================

list=c("tidyverse","stringr","forcats","ggmap","rvest","tm","SnowballC","dplyr","calibrate","doParallel",
       "stringi","ggplot2","maps","httr","rsdmx","devtools","plyr","dplyr","ggplot2","caret","elasticnet",
       "magrittr","broom","glmnet","Hmisc",'knitr',"RSQLite","RANN","lubridate","ggvis","plotly","lars",
       "ggcorrplot","GGally","ROCR","lattice","car","corrgram","ggcorrplot","sqldf","parallel")

list_packages <- list
new.packages <- list_packages[!(list_packages %in% installed.packages()[,"Package"])]
if(length(new.packages)) install.packages(new.packages)

sapply(list, require, character.only = TRUE)


```

```{r}
#==================================================================
#  Set up parallel processing
# leave two cores for operating system
#==================================================================

cluster <- makeCluster(detectCores() - 2) 
registerDoParallel(cluster)

```



The  September 2015 data green taxi  contains 1048575  rows  and      21 columns.


```{r}
#==================================================================
# load the downloaded data with the readr package
#  dim function produces the dimension of the data
# the  September 2015 data contains 1048575  rows  and      21 columns
#==================================================================

data=readr::read_csv("C:/Users/Gucci148/Documents/DataMiningscience/Capitalone/green_tripdata_2015-09.csv")

names(data)

dim(data)
```










```{r}
#==================================================================
# Look at  the  structure of the data with the glimpse function in 
#  dplyr  package
#==================================================================

dplyr::glimpse(data)

summary(data)


```

```{r}
#==================================================================
# 
#  1 check missing obervations of each column variable
#  2 sum of all missing obervations in data
#  3   Ehail_fee  has all rows missing,remove from dataset
#==================================================================



apply(is.na.data.frame(data),2,sum)


newdata=dplyr::select(data,-Ehail_fee)

sum(is.na(newdata))
```



The average elapsed trip distance  reported by the taximeter is approximately 3 miles The median elapsed trip distance  reported by the taximeter is approximately 2 miles. The highest  elapsed trip distance  reported by the taximeter is approximately 603 miles and lowest is 0. The 603 mile appears to be an outlier. The data is skewed to the right with most observations are between zero and four.
```{r}
#==================================================================
# 
# Histogram of the Distribution of Trip Distance
# with ggplot2 package
#==================================================================

p<-ggplot(newdata, aes(x =Trip_distance )) + 
  geom_histogram(fill="black",col="black",alpha=0.2,binwidth=0.02) + 
theme_minimal() + ggtitle(" Histogram of the Distribution of Trip Distance") +
  #center title
  theme(plot.title = element_text(hjust = 0.5)) +xlab("Trip Distance")+ylab("Frequency")


ggplotly(p)
```



Right skewed distributions are not symmetric and transformations is usually done in order to build any model that uses a normal distribution assumption.Transformations improve symmetry,linear  relationships and constant variance which are usually need for models such as regression. A common transformation of right skewed data is the log transformation. 

```{r}

#==================================================================
# 
# Histogram of the Distribution of Log Trip Distance
# with ggplot2 package
#==================================================================

newdata=newdata%>%mutate(New_distance=log(Trip_distance+1)) 

p=ggplot(newdata, aes(x =New_distance ))+  geom_histogram(fill="black",col="black",binwidth=0.5) +
  theme_minimal() + ggtitle(" Histogram of the Distribution of Trip Distance") +
  #center title
  theme(plot.title = element_text(hjust = 0.5)) +xlab("Log of Trip Distance")+ylab("Frequency")

ggplotly(p)


```


The summary statistics of the transformed Trip distance shows the mean is about 3.027 and median is 2.010.


```{r}
newdata%>%mutate(Transform=log(Trip_distance+1))%>%dplyr::select(Trip_distance)%>%summary%>%knitr::kable()
```

The mean and median trip distance grouped by hour of day. The maximum mean  and median trip distance by hour 
 which is 4.192 and 2.95 respectively was travelled at fifth hour.

```{r}
#===========================================================================
# 
# Mean and median trip distance grouped by hour of day.
# Create Hours variable by extracting hours from the 
#  the  date and time when the meter was engaged (lpep_pickup_datetime).
#===========================================================================

newdata=newdata%>%dplyr::mutate(Hours= format(as.POSIXct(strptime(newdata$lpep_pickup_datetime,"%m/%d/%Y %H:%M",tz="")) ,format = "%H"))

newdata%>% mutate(Hours=as.numeric(Hours))%>%group_by(Hours)%>% dplyr::summarise(Mean=mean(Trip_distance),Median=median(Trip_distance),n=n())


```





```{r}
#===========================================================================
# Find the Hour in which  the  maximum mean  Trip distance occured
#===========================================================================

newdata%>% mutate(Hours=as.numeric(Hours))%>%group_by(Hours)%>% dplyr::summarise(Mean=mean(Trip_distance),Median=median(Trip_distance),n=n())%>%
  dplyr::slice(which.max(Mean )) 
```




```{r}

#===========================================================================
# Find the Hour in which  the  maximum  median  Trip distance occured
#===========================================================================

newdata%>% mutate(Hours=as.numeric(Hours))%>%group_by(Hours)%>% dplyr::summarise(Mean=mean(Trip_distance),Median=median(Trip_distance),n=n())%>%
  dplyr::slice(which.max(Median )) 
```





The new variable Tip is created by dividing Tip amount by the Total amount.The minimum Tip amount is -12.30. This appears to 
be an error in the data. The Tip amounts distribution is also right skewed  with most observations between 0 and 10. (99%).



```{r}
#==================================================================
# 
# derived variable for tip as a percentage of the total fare
#==================================================================


newdata=newdata%>%dplyr::mutate(Tip=(Tip_amount/Total_amount)*100)

length(newdata$Tip_amount[newdata$Tip_amount<10])/length(newdata$Tip_amount)
```

```{r}
#==================================================================
# 
# Histogram of the Distribution of Tip_amount
# with ggplot2 package
#==================================================================

p<-ggplot2::ggplot(newdata, aes(x =Tip_amount )) + 
  geom_histogram(fill="black",col="black",alpha=0.2,binwidth=0.02) + 
theme_minimal() + ggtitle(" Histogram of the Distribution of Tip amount") +
  #center title
  theme(plot.title = element_text(hjust = 0.5)) +xlab("Tip_amount")+ylab("Frequency")

ggplotly(p)


```

```{r}
# log transformation ,13.3 is added to prevent a case of taking log(0)

newdata=newdata%>%dplyr::mutate(New_Tip=log(Tip_amount+13.3))
                                       
```


The log transformation did not help out much in moving the data close to normal.We keep the old form in further analysis.
```{r}
p=ggplot2::ggplot(newdata, aes(x =New_Tip )) + 
  geom_histogram(fill="black",col="black",alpha=0.2,binwidth=0.2) + 
theme_minimal() + ggtitle(" Histogram of the Distribution of New Tip amount") +
  #center title
  theme(plot.title = element_text(hjust = 0.5)) +xlab("New Tip amount")+ylab("Frequency")

ggplotly(p)

```





```{r}

#==============================================================================
#   
# Building a Predictive Model
# Preprocessing data
# 1 There are three missing variables that will be removed at preprocessing step
# 2  Center and scale numerical columns to reduce variability
#==============================================================================


newdata=newdata[complete.cases(newdata),]

sum(is.na(newdata))


summary(newdata)

```




```{r}
#==================================================================
#Spliting training set into two parts based on outcome:
# Training set which 70% and Test set which 30% of the data
#==================================================================


index <- createDataPartition(newdata$Tip_amount, p=0.70, list=FALSE)

trainSet <- newdata[ index,]

testSet <- newdata[-index,]


sum(is.na(trainSet))
sum(is.na(testSet))
sum(is.na(newdata))

```


There are very  high positive  correlations between Total amount and Trip distance,of course which makes sense 
Total amount and Fare amount that passengers who travelled longer distances  were charged more.
There is also a significant positive correlation between Total amount and Tip amount.
Passengers who travelled longer distances and paid higher fares were more likely to tip.

```{r}
#==================================================================
#  Exploratory data analysis
# Determine correlation between variables
#==================================================================




newdata%>%dplyr::select(New_distance,Fare_amount,improvement_surcharge,Tolls_amount,Tip_amount,MTA_tax
                        ,Extra,Passenger_count,VendorID, Tip,Total_amount)%>%cor



```




```{r}
ggplot(trainSet , aes(Tip, Tip_amount))+geom_point(color="purple")+
  ggtitle('Tip vs Tip_amount')+
  xlab('Tip')+ylab('Tip_amount')+
  stat_smooth(method=lm, colour='black',span=0.2)
```
There exist a strong association between Tip and Tip amount.




```{r}
ggplot(trainSet , aes(Tip, Fare_amount))+geom_point(color="purple")+
  ggtitle('Tip vs Fare_amount')+
  xlab('Tip')+ylab('Fare_amount')+
  stat_smooth(method=lm, colour='black',span=0.2)
```





```{r}
ggplot(trainSet , aes(Tip, Trip_distance))+geom_point(color="purple")+
  ggtitle('Tip vs Trip_distance')+
  xlab('Tip')+ylab('Trip_distance')+
  stat_smooth(method=lm, colour='black',span=0.2)
```

The figure shows there are a few outlier points ,where the Tip was low for a very long Trip distance



```{r}
ggplot(trainSet , aes(Tip, New_distance))+geom_point(color="purple")+
  ggtitle('Tip vs  New_distance')+
  xlab('Tip')+ylab('New_distance')+
  stat_smooth(method=lm, colour='black',span=0.2)
```



```{r}
ggplot(trainSet , aes(Tip, Tolls_amount))+geom_point(color="purple")+
  ggtitle('Tip vs Tolls_amount')+
  xlab('Tip')+ylab('Tolls_amount')+
  stat_smooth(method=lm, colour='black',span=0.2)
```


 The Figure does reveal a weak positive association  between Tolls amount and Tip given.


```{r}
#==================================================================
#  Do feature selection using Generalized Linear Model
#==================================================================



control <- trainControl(method="repeatedcv", number=10, repeats=5)

my_glm<- train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax","improvement_surcharge",
                          "Extra","Passenger_count","VendorID")], trainSet$Tip,
                    method = "glm",
                    preProc = c("center", "scale"),
                    trControl = control)

p=ggplot(varImp(my_glm, scale = FALSE))+ggtitle("Variable Importance using GLM")+theme_minimal()

ggplotly(p)
```

From the variable importance plot, we drop can drop improvement surcharge  since it has  have very 
little predictive power for Tip.


```{r}
#===============================================================================
#  Build a  Generalized Linear Model For with Stepwise Selection prediction
#===============================================================================

control <- trainControl(method="repeatedcv", number=10, repeats=5)


step_glm<- train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                          "Extra","Passenger_count","VendorID")], trainSet$Tip,
                    method = "glmStepAIC",
                    preProc = c("center", "scale"),
                    trControl = control)

```

```{r}
pred<-predict(step_glm,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                          "Extra","VendorID")])


glm_data=data_frame(predicted=pred,observed=testSet$Tip)

ggplot(glm_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('GLM')

```


```{r}
# Print, plot variable importance

print(varImp(my_glm, scale = FALSE))


```



```{r}
plot(varImp(my_glm, scale = FALSE), main="Variable Importance using GLM")


```


```{r}
# Mean Squared Error

sqrt(mean(pred- testSet$Tip)^2)
```




```{r}
#===============================================================================
#  Build a  Partial Least Squares Model 
#===============================================================================

# Partial Least Squares is one way to reduce dimension of the predictors used in the model. It identifies
#linear combinations,or directions, that best represent the predictors in the data.The directions are identified in 
# unsupervised way since the outcome variable is not in identifying the principal
# component directions.The predictors are preprocessed by centering and scaling.
# PLS will seek
#directions of maximum variation while simultaneously considering correlation with the response.


control <- trainControl(method="repeatedcv", number=10, repeats=5)

plsfit=train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                  "Extra","VendorID","Passenger_count")], trainSet$Tip,
      method = "glmStepAIC",tuneLength = 20,
      preProc = c("center", "scale"),
      trControl = control)






# Predict using test data


pred<-predict(plsfit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                            "Extra","VendorID")])

my_data=data_frame(predicted=pred,observed=testSet$Tip)

ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('PLS')

```


```{r}
# Print, plot variable importance
print(varImp(plsfit, scale = FALSE))

```


```{r}
plot(varImp(plsfit, scale = FALSE), main="Variable Importance using PLS")
```


```{r}
# Mean Squared Error

sqrt(mean(pred- testSet$Tip)^2)
```




```{r}
#===============================================================================
#  Build Penalized linear regression models Model 
# Lasso (least absolute shrinkage and selection operator)
#===============================================================================

# The LASSO  penalizes the model  for having many predictors by shrinking the coefficients of 
# 
# those predictors with little variation to zero thereby reducing the dimension of the model.

lassoGrid <- expand.grid(.fraction = seq(.05, 1, length = 20))

control <- trainControl(method="repeatedcv", number=10, repeats=5)

set.seed(100)




lassofit=train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                  "Extra","VendorID")], trainSet$Tip,
      method = "lasso",tuneGrid = lassoGrid,
      preProc = c("center", "scale"),
      trControl = control)


# Predict using test data
pred<-predict(lassofit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                                  "Extra","VendorID")])

my_data=data_frame(predicted=pred,observed=testSet$Tip)

ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('LASSO Model')

```

```{r}
print(varImp(lassofit, scale = FALSE), main="Variable Importance using lasso")
```


```{r}
plot(varImp(lassofit, scale = FALSE), main="Variable Importance using lasso")
```



```{r}
plot(lassofit)
```


```{r}
#Extract coefficients of final model

predict.enet(lassofit$finalModel, type='coefficients', s=lassofit$bestTune$fraction, mode='fraction')

```



```{r}
# Mean Squared Error

sqrt(mean(pred- testSet$Tip)^2)
```


I ran out memory after  several attempts to fit a random forest model.
```{r}
#===============================================================================
#  Random Forest linear regression models Model 
# 
#===============================================================================



control <- trainControl(method="repeatedcv", number=10, repeats=5)



rf_fit<-train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                  "Extra","VendorID")], trainSet$Tip,
      method = "rf", ntrees = 1000,importance = TRUE,
      preProc = c("center", "scale"),
      trControl = control)

# Predict using the test data

pred<-predict(rf_fit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                                  "Extra","VendorID")])

my_data=adata_frame(predicted=pred,observed=testSet$Tip)

ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('Random Forest Model')

```


```{r}
plot(varImp(rf_fit, scale = FALSE),main="Variable Importance using RF")
```



```{r}
# Mean Squared Error

sqrt(mean(pred- testSet$Tip)^2)
```


I ran out memory after  several attempts to fit a gradient boosting machine.
```{r}
#===============================================================================
# Gradient Boosting Machine  Model 
# 
#===============================================================================

# To tune over interaction depth, number of trees, and shrinkage first define a tuning grid,
#then train over this grid


gbmGrid <- expand.grid(.interaction.depth = c(2,5,8),
                       .n.trees = c(500, 1000,2000,5000),
                       .shrinkage = c(0.01, 0.1),
                       .n.minobsinnode=c(5,10,15))

control <- trainControl(method="repeatedcv", number=10, repeats=5)


gbm_fit<-train(trainSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                    "Extra","VendorID")], trainSet$Tip,
        method = "gbm",verbose=F,tuneGrid = gbmGrid,
        preProc = c("center", "scale"),
        trControl = control)

# Predict using the test data

pred<-predict(gbm_fit,testSet[,c("Fare_amount","Trip_distance","Tolls_amount","MTA_tax",
                                "Extra","VendorID")])

my_data=adata_frame(predicted=pred,observed=testSet$Tip)


ggplot(my_data,aes(predicted,observed))+geom_point()+geom_smooth(method=lm)+ggtitle('Boosting')




```



```{r}
plot(varImp(gbm_fit, scale = FALSE),main="Variable Importance using Boosting")
```



```{r}
# Mean Squared Error

sqrt(mean(pred- testSet$Tip)^2)
```



We constructed a derived variable of time difference by subtracting pick up time from drop off time.The average speed of any 
trip was obtained by dividing distance covered by the time take. The time is converted to seconds to hours
which is standard  form.The Overall average speed off all trips is about 13 miles per hour.

```{r}
#Build a derived variable representing the average speed over the course of a trip.
#Average Speed=(distance)/time

# 1 create a TimeDifference variable by subtracting  lpep_pickup_datetime
# from Lpep_dropoff_datetime using the difftime function.
#2 Extract the  numeric values  in difference  in time(in seconds) by 
# by the stri_extract_all_regex function
# 3 unlist the result which is characer and convert to numeric
# 4  convert time  in seconds  to hours by multiplying by 0.000277778
# This standardizes speed in scientific units which is miles per hour
# 5 Time difference greater than zero seconds were selected in computing the 
#  speed to avoid undefined mathematical expressions
# 6 Speed variable=Trip distance/ TimeDifference

# Average speed=mean of the Speed variable

# The Average speed over all of september is approximately 5.844  meters per second


newdata=newdata%>%mutate(TimeDifference=
              as.numeric(unlist( stri_extract_all_regex(difftime(as.POSIXct(Lpep_dropoff_datetime, format="%m/%d/%Y %H:%M", tz="")
                                                                 , as.POSIXct(lpep_pickup_datetime, format="%m/%d/%Y %H:%M", tz="")
                                                                 , tz="",units = , "secs"), "[0-9]+"))))%>%mutate(Time=TimeDifference*0.000277778)



newdata=newdata[newdata$Time>0,]



#newdata=newdata%>%mutate(Speed=Trip_distance/TimeDifference)%>%dplyr::filter[newdata,TimeDifference >0]

# The filter approach seems not to recognise TimeDifference variable just created.

newdata=newdata%>%mutate(Speed=Trip_distance/Time)




```



The overall average speed for all  trips by green taxis is found to be 13.06 miles per hour.

```{r}
newdata%>%dplyr::summarise(Average_Speed=mean(Speed))

```

 The average trip speeds are materially different  in all weeks of September.We tested for the difference in 
 means between the three weeks in September per the dataset by an using Kruskal-Wallis and ANOVA model.Since the trip distance is right skewed,we proceed further to use a non-parametric test like Kruskal-Wallis to  test the differences between the speeds over the three weeks in September.Both parametric Anova without any transformations gives a similar outcome 
 like the non-parametric Kruskal-Wallis test.We found thet  there is  asignificant difference in the average Speed  by week travelled in September(p-value  < 2.2e-16).The first week had the highest average speed of about 13.75.
 Labor weekend occurred in the  first week of Septtember 2015,that could have contributed to the
 Higher speed in the first week of September as a lot of people were in a rush to get to their destinations
 For the holidays.

```{r}
#  Test of the mean and Median of average trip speeds
# in all weeks of September
# 1 Create a derived variable Week for the number of weeks in Septemeber
#  present in the data
# 2  We then group speed by week 
# 3 Test for difference in Mean speed for the weeks in September.
# 4 We will perform Kruskal Wallis nonparametric test and compare with parametric ANOVA 
#test of  the mean Speeds in the 3 weeks in September. The ANOVA test assumes the data is
# normally distributed wheras Kruskal-Wallis test does not.

week1=c("09/01/2015","09/02/2015","09/03/2015","09/04/2015","09/05/2015","09/06/2015","09/07/2015")
week2=c("09/08/2015","09/09/2015","09/10/2015","09/11/2015","09/12/2015","09/13/2015","09/14/2015")
week3=c("09/15/2015","09/16/2015","09/17/2015","09/18/2015","09/19/2015","09/20/2015","09/21/2015")

newdata=newdata%>%
  dplyr::mutate(dates= format(as.POSIXct(strptime(newdata$lpep_pickup_datetime,"%m/%d/%Y %H:%M",tz="")) ,format = "%m/%d/%Y"))

newdata=newdata %>%  
  mutate(Week = case_when(.$dates %in% week1 ~ 1, .$dates %in% week2 ~ 2,.$dates %in% week3 ~ 3)) 

newdata
```



```{r}
# Find the mean speed in each in September

newdata %>%group_by(Week)%>% dplyr::summarise(mean=mean(Speed))
 
```


```{r}
newdata %>%dplyr::select( Week,Speed)%>%mutate(Week=factor(Week))%>%kruskal.test()%>%tidy()
```



```{r}
newdata %>%dplyr::select( Week,Speed)%>%lm(formula = Speed ~ factor(Week ))%>%anova()%>%tidy()

```



A plausible Hypothesis would be to test if speed is associated with time of the day.This can be done by building a Linear regression model.
 There exist a significant association between Speed and Hour of the day. The coefficient of Hours is negative which indicates a negative relationship between the two variables. For every unit increase in Hour of the day,the 
 Speed decreases by a factor of about 0.106343.


```{r}
ggplot(newdata, aes(Hours, Speed))+geom_point(color="purple")+
  ggtitle('Hours vs Speed')+
  xlab('Hours')+ylab('Speed')+
  stat_smooth(method=lm, colour='black',span=0.2)

```


```{r}
newdata %>%dplyr::select(Speed,Hours)%>%mutate( Hours=as.numeric( Hours))%>%
  lm(formula = Speed ~ (Hours ))%>%summary()


```


```{r}
fit_glm=glm( Speed ~ as.numeric(Hours), data=newdata)


tidy(fit_glm)
```


```{r}
head(augment(fit_glm))
```



```{r}
glance(fit_glm)
```





Other insights one would consider given ample time include the following:

1. Spatial data analysis with get_map and ggmap to determine if the green taxis followed the citys order to pick up 
paasengers from Manhattan eclusionary zone.
2. Building predictive models with deep neural networks which is known to have better predictive power.

3. Showing interactive plots to assist in detecting any underlying patterns of the data.